US11041801B2 - Method for estimating a quantity of a gaseous species - Google Patents
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- US11041801B2 US11041801B2 US16/622,703 US201816622703A US11041801B2 US 11041801 B2 US11041801 B2 US 11041801B2 US 201816622703 A US201816622703 A US 201816622703A US 11041801 B2 US11041801 B2 US 11041801B2
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
- G01N21/00—Investigating or analysing materials by the use of optical means, i.e. using sub-millimetre waves, infrared, visible or ultraviolet light
- G01N21/17—Systems in which incident light is modified in accordance with the properties of the material investigated
- G01N21/25—Colour; Spectral properties, i.e. comparison of effect of material on the light at two or more different wavelengths or wavelength bands
- G01N21/31—Investigating relative effect of material at wavelengths characteristic of specific elements or molecules, e.g. atomic absorption spectrometry
- G01N21/35—Investigating relative effect of material at wavelengths characteristic of specific elements or molecules, e.g. atomic absorption spectrometry using infrared light
- G01N21/3504—Investigating relative effect of material at wavelengths characteristic of specific elements or molecules, e.g. atomic absorption spectrometry using infrared light for analysing gases, e.g. multi-gas analysis
-
- G—PHYSICS
- G01—MEASURING; TESTING
- G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
- G01N21/00—Investigating or analysing materials by the use of optical means, i.e. using sub-millimetre waves, infrared, visible or ultraviolet light
- G01N21/17—Systems in which incident light is modified in accordance with the properties of the material investigated
- G01N21/25—Colour; Spectral properties, i.e. comparison of effect of material on the light at two or more different wavelengths or wavelength bands
- G01N21/31—Investigating relative effect of material at wavelengths characteristic of specific elements or molecules, e.g. atomic absorption spectrometry
- G01N21/314—Investigating relative effect of material at wavelengths characteristic of specific elements or molecules, e.g. atomic absorption spectrometry with comparison of measurements at specific and non-specific wavelengths
-
- G—PHYSICS
- G01—MEASURING; TESTING
- G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
- G01N21/00—Investigating or analysing materials by the use of optical means, i.e. using sub-millimetre waves, infrared, visible or ultraviolet light
- G01N21/17—Systems in which incident light is modified in accordance with the properties of the material investigated
- G01N21/25—Colour; Spectral properties, i.e. comparison of effect of material on the light at two or more different wavelengths or wavelength bands
- G01N21/27—Colour; Spectral properties, i.e. comparison of effect of material on the light at two or more different wavelengths or wavelength bands using photo-electric detection ; circuits for computing concentration
- G01N21/274—Calibration, base line adjustment, drift correction
-
- G—PHYSICS
- G01—MEASURING; TESTING
- G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
- G01N2201/00—Features of devices classified in G01N21/00
- G01N2201/12—Circuits of general importance; Signal processing
- G01N2201/121—Correction signals
Definitions
- the technical field of the present disclosure is the use of a light source, in particular a black body or gray body, to perform optical measurements, with a temporal drift affecting the light radiation emitted by the light source taken into account.
- Optical methods are frequently used to analyze a gas. Some devices allow the composition of a gas to be determined on the basis of the fact that the species from which the gas is composed have absorption spectral properties that are different from one another. Thus, if the absorption spectral band of a gaseous species is known, its concentration may be determined via an estimation of the absorption of light passing through the gas, using the Beer-Lambert law. This principle allows the concentration of a gaseous species present in the gas to be estimated.
- the light source is usually a source emitting in the infrared, the method used conventionally being referred to as NDIR detection, the acronym NDIR meaning “non-dispersive infrared.”
- NDIR detection the acronym NDIR meaning “non-dispersive infrared.”
- the analyzed gas lies between a light source and a photodetector, called the measurement photodetector, the latter being intended to measure a light wave transmitted by the gas to be analyzed, the light wave being partially absorbed by the latter.
- These methods generally comprise a measurement of a light wave, called the reference light wave, emitted by the source, the reference light wave not being absorbed, or being absorbed negligibly, by the analyzed gas.
- Comparison of the light wave in the presence of gas and the light wave without gas allows the absorption of the gas to be characterized. It is, for example, a question of determining an amount of a gaseous species in the gas, using the technology referred to as “absorption NDIR.” It may also be a question of estimating a number of particles in the gas, by detecting light scattered by the latter in a preset angular range of scatter.
- the reference light wave is measured by a reference photodetector. It may be a reference photodetector different from the measurement photodetector, and arranged so as to be placed facing the light source, the reference photodetector being associated with a reference optical filter.
- the reference optical filter defines a reference spectral band, in which the gas to be analyzed exhibits no significant absorption.
- the reference photodetector is frequently affected by measurement noise, which has an impact on the estimation of the intensity of the reference light wave. This is, in particular, the case when the reference photodetector is a simple and inexpensive photodetector. Such noise may lead to uncertainty in the estimated amounts.
- Embodiments of the present disclosure aim to decrease this uncertainty, by limiting the effect of the fluctuations affecting the measurements carried out by the reference photodetector.
- a first subject of the present disclosure is a method for measuring an amount of a gaseous species present in a gas, the gaseous species being able to absorb light in an absorption spectral band, the method comprising the following steps:
- steps b) to d) being implemented at a plurality of measurement times, the method also comprising the following steps:
- step e) comprises the following substeps:
- the reference light wave is representative of a light wave that reaches the measurement photodetector without having been absorbed by the gas. It lies in a reference spectral band. Depending on the configuration, the reference spectral band may be separate from or none other than the absorption spectral band.
- the model taken into account may, in particular, be a linear model, step eii) forming a linear regression.
- steps b) to d) are carried out at various measurement times forming a time range, and, following these measurement times:
- each time of the time range may be assigned a weighting term that is strictly positive and lower than or equal to 1.
- sub steps eii) and eiii) are implemented at each measurement time, iteratively, the parameters of the model being updated depending on parameters of the model resulting from a preceding iteration, or, at a first measurement time, depending on initialized parameters.
- steps e) and f) may be implemented at each measurement time.
- Sub step eii) may take into account a forgetting factor to weight to what extent the preceding iteration is taken into account.
- the model may comprise a weighting term that is strictly positive and lower than or equal to 1.
- step f) may comprise the following sub steps:
- Another subject of the present disclosure is a device for determining an amount of a gaseous species in a gas, the device comprising:
- the first and second processors may be merged and form a single processor.
- FIG. 1 shows a device allowing the present disclosure to be implemented.
- FIGS. 2A and 2B illustrate a variation in the reference light intensities and measurement light intensities measured by a reference photodetector and a measurement photodetector of a device, such as shown in FIG. 1 , respectively, in various configurations.
- FIG. 2C shows a detail of FIG. 2B .
- FIG. 3 shows an example of correction of the light intensity emitted by the light source.
- FIGS. 4A and 4B show the main steps of one embodiment of the present disclosure.
- FIGS. 4C and 4D show the main steps of another embodiment of the present disclosure.
- FIGS. 5A, 5B and 5C show estimations, called denoised estimations, of the reference intensities measured by a reference photodetector, in three embodiments.
- FIG. 1 is an example of a device 1 for analyzing gas.
- This device comprises a chamber 10 defining an internal space inside of which are located:
- the gas G contains a gaseous species G x an amount c x (k) of which, a concentration of which, for example, it is sought to determine at a measurement time k.
- This gaseous species absorbs a measurable percentage of the light in an absorption spectral band ⁇ x .
- the light source 11 is able to emit the incident light wave 12 , in an illumination spectral band ⁇ , the latter possibly lying between the near ultraviolet and the mid infrared, between 200 nm and 10 ⁇ m, and most often between 1 ⁇ m and 10 ⁇ m.
- the absorption spectral band ⁇ x of the analyzed gaseous species is comprised in the illumination spectral band ⁇ .
- the light source 11 may, in particular, be pulsed, the incident light wave 12 being a pulse of duration generally comprised between 100 ms and 1 s.
- the light source 11 may, in particular, be a suspended filament light source heated to a temperature comprised between 400° C. and 800° C.
- the measurement photodetector 20 is preferably associated with an optical filter 18 , defining a detection spectral band encompassing all or some of the absorption spectral band ⁇ x of the gaseous species.
- the measurement photodetector 20 is a thermopile, able to deliver a signal dependent on the intensity of the light wave to which the photodetector is exposed. It may also be a question of a photodiode or of another type of photodetector.
- the intensity I(k) of the light wave 14 detected by the measurement photodetector 20 depends on the amount c x (k) at the measurement time, according to the Beer-Lambert equation:
- I ⁇ ( k ) I x ⁇ ( k ) corresponds to an attenuation att(k) generated by the gaseous species in question at the time k.
- Expression (1) assumes control of the intensity I x (k) of the incident light wave 12 at the measurement time k.
- the device comprises a reference photodetector 20 ref , arranged such that it detects a light wave, called the reference light wave 12 ref , representative of the incident light wave 12 emitted by the light source 11 .
- the reference light wave 12 ref reaches the reference photodetector 20 ref without interacting with the gas G, or without significantly interacting with the latter.
- the intensity of the reference light wave 12 ref detected by the reference photodetector 20 ref , at the measurement time k, is referred to by the term reference intensity I ref (k).
- the reference light wave lies in a reference spectral band ⁇ ref .
- the reference photodetector 20 ref is placed beside the measurement photodetector 20 and is of the same type as the latter. It is associated with an optical filter, called the reference optical filter 18 ref .
- the reference optical filter 18 ref defines the reference spectral band ⁇ ref corresponding to a range of wavelengths not absorbed by the gaseous species in question.
- the reference spectral band ⁇ ref is, for example, centered on the wavelength 3.91 ⁇ m.
- I ref (k) In prior-art devices, measurement of I ref (k) allows expression (1) to be used with I x (k) estimated from I ref (k), this allowing ⁇ (c x (k)) to be determined, then c x (k) to be estimated.
- the reference photodetector 20 ref may be affected by a large amount of read noise, impacting the precision of the determination of the reference intensity I ref (k).
- the reference intensity is thus subject to statistical fluctuations, this resulting in a high measurement uncertainty that has an impact on the estimation of the amount c x (k) of the gaseous species.
- the present disclosure addresses this problem, by correcting the reference intensity I ref (k) measured at each measurement time. More precisely, the correction consists in replacing the reference intensity I ref (k), measured at each measurement time, with an estimation I′ ref (k) of the reference intensity called the denoised estimation.
- I′ ref (k) corresponds to the corrected reference intensity.
- the device comprises a first processor 30 , for example, a microprocessor or a microcontroller.
- the latter is configured to receive a signal representative of the intensity I ref (k) of the reference light wave 12 ref , measured by the reference photodetector 20 ref at each measurement time k, and to implement a method in order to obtain a corrected reference intensity I′ ref (k).
- the correcting method is described below, with reference to FIGS. 4A to 4D .
- the first processor 30 is connected to a memory 32 containing instructions allowing certain steps of this method to be implemented.
- the first and second processors may be one and the same processor.
- the device also comprises a second processor 30 ′ configured to receive a signal representative of a measurement intensity I(k) and the corrected reference intensity I′ ref (k).
- the second processor is programmed to determine, depending on these intensities, the amount of the gaseous species measured at each measurement time.
- the optical filter 18 defined a detection spectral band of 160 nm width about the wavelength 4.26 ⁇ m.
- Four devices were used, using light sources raised to different potentials V1, V2, V3 and V4, with V1>V2>V3>V4, respectively.
- the variation, over time, in the reference intensity I ref and in the measurement intensity I were measured.
- the index k is a temporal index, and designates a light pulse emitted at a measurement time k.
- each light pulse had a duration of 150 ms, the pulse frequency being one pulse every 300 ms, i.e., 1/0.3 Hz.
- the measurement photodetector 20 and the reference photodetector 20 ref were formed by a Heimann Sensor thermopile of reference HCM Cx2 Fx. The measured intensity of each pulse is here expressed in units of voltage, corresponding to the voltage across the terminals of the thermopile.
- FIG. 2B shows the temporal variation in the measurement intensity I detected, in each configuration, by the measurement photodetector 20 .
- These trials were carried out for the most part by analyzing ambient air, the carbon-dioxide concentration being the ambient concentration C amb .
- a few series of point measurements were carried out while the carbon-dioxide concentration was varied. These point measurements resulted in the fluctuations that may be seen in the curves of FIG. 2B .
- One of these fluctuations has been encircled by a dashed line and is detailed in FIG. 2C .
- FIG. 2C shows the series of measurements shown in FIG.
- the concentration passed from ambient concentration C amb (about 400 ppm) to C0 the incident light wave 12 was less attenuated, this resulting in an increase in the measurement intensity I.
- the measurement intensity delivered by the measurement photodetector corresponds to curve I of this figure.
- the statistical fluctuations may be decreased by applying a median filter, so as to obtain a filtered intensity I f allowing the useful information to be preserved while attenuating the main fluctuations in the measured intensity.
- the gradual decrease in the measurement intensity may be corrected by taking into account the reference intensity.
- a compensation function comp may thus be applied to the measurement intensity, preferably after application of a median filter, such that, at each measurement time k,
- I ref,f being the reference intensity after application of a median filter to five successive samples.
- the inventors have sought to optimize the way in which the reference intensity is taken into account, so as to further limit the fluctuations affecting the latter. They have taken advantage of the fact that, contrary to the measurement intensity, certain fluctuations of which are due to non-modellable variations in concentrations of the analyzed gaseous species, the variation in the reference intensity may be modelled using a predetermined parametric model. If ⁇ is a vector containing the parameters of the model, determining ⁇ allows a denoised estimation of the reference intensity I ref to be obtained.
- FIGS. 4A and 4B A first embodiment of the present disclosure is schematically shown in FIGS. 4A and 4B .
- Step 100 selecting the model.
- a parametric model is selected.
- the measurement intensity I(k) and the reference intensity I ref (k) are acquired at each measurement time k.
- Step 120 reiterating step 110 or stopping the iteration.
- Step 110 is reiterated until a number N k of iterations has been reached.
- the decrease in the time range ⁇ k employed to establish the vector of parameters ⁇ may allow the uncertainty in the model with respect to the measurements to be decreased, as described below.
- Step 130 estimating the vector of parameters ⁇ circumflex over ( ⁇ ) ⁇ .
- the first column is formed by all of the time increments k in increasing order, the second column being formed from 1's.
- the vector of parameters ⁇ circumflex over ( ⁇ ) ⁇ may be estimated by minimizing the quadratic norm of the error vector ⁇ , this being expressible by the following expression:
- This estimation is carried out in substep 135 .
- Step 140 Correcting the reference intensity.
- the model is taken into account to correct the reference intensity I ref (k), so as to obtain a corrected reference intensity I′ ref (k).
- Step 150 Estimating the amount c x (k) of the gaseous species analyzed.
- This estimation is carried out by estimating, from I′ ref (k), the intensity I x (k) reaching the measurement photodetector 20 in the absence of gas, in the absorption spectral band ⁇ x .
- the computation of the ratio is carried out by estimating, from I′ ref (k), the intensity I x (k) reaching the measurement photodetector 20 in the absence of gas, in the absorption spectral band ⁇ x .
- I ⁇ ( k ) I x ⁇ ( k ) allows the amount c x (k) to be obtained as indicated above.
- FIG. 5A shows, for various potentials to which the light source was raised, the measured reference intensity I ref and the reference intensity I′ ref corrected using the method described with regard to steps 100 to 150 .
- the corrected reference intensity I′ ref is not subject to statistical fluctuations.
- the curves corresponding to the potentials V2, V3 and V4, respectively it describes with fidelity the temporal variation in the reference intensity I ref .
- the correction may be improved by decreasing the time range ⁇ k taken into account to establish the model.
- the reference intensity is then approximated by a piecewise linear model, the duration of each piece corresponding to a time range ⁇ k. Each piece may then form the subject of an estimation of a vector of parameters, according to steps 100 to 150 .
- the precision of the model may be improved by taking into account a weighting factor, or forgetting factor ⁇ , associated with each time increment k, with ⁇ ]0,1], the forgetting factor preferably being comprised between 0 and 1.
- the forgetting factor in question allows a weighting matrix W, of (N k , N k ) size, to be formed such that:
- equation (7) is equivalent to equation (5).
- Step 200 Selecting the model.
- step 100 is similar to step 100 described above.
- Step 210 acquiring a measurement.
- the measurement intensity I(k) and the reference intensity I ref (k) are acquired at a measurement time k.
- Step 220 Updating the vector of parameters ⁇ k corresponding to the measurement time k.
- Each vector ⁇ k is such that its transpose ⁇ k T corresponds to the k th row of the observation matrix ⁇ described above with regard to step 130 ;
- ⁇ k [ 1 1 ⁇ ⁇ k 1 ]
- P k ( ⁇ k T ⁇ k ) ⁇ 1 .
- P k corresponds to the inverse of the autocorrelation matrix of the observation matrix ⁇ k .
- P k and ⁇ k are quantities that are updated at each measurement time k and that allow the estimation ⁇ circumflex over ( ⁇ ) ⁇ k of the vector of parameters of the model at the time.
- This embodiment is particularly advantageous because it allows the matrix P k to be expressed as a function of P k-1 , the expression of this matrix in a preceding iteration k ⁇ 1, using equation (10). This does not require the matrix inversion that the preceding embodiment requires (see equation (5)).
- expression (9) is implemented considering an arbitrary initial vector of parameters ⁇ circumflex over ( ⁇ ) ⁇ 0 , for example
- Step 220 is illustrated in FIG. 4D . It comprises:
- the model parameterized in step 220 is taken into account to correct the reference intensity I ref (k), so as to obtain a corrected reference intensity I′ ref (k).
- Step 240 Estimating the amount c x (k) of the gaseous species analyzed.
- This embodiment is said to be recursive because it uses quantities P k-1 and ⁇ circumflex over ( ⁇ ) ⁇ k-1 obtained from the preceding iteration or, in the first iteration, initialized quantities.
- the vector of parameters ⁇ circumflex over ( ⁇ ) ⁇ k is updated on each iteration, this allowing, in each iteration, it to be taken into account to correct the reference intensity I ref (k) and to obtain the concentration c x (k) of the sought-after gaseous species.
- a weighting term, or forgetting factor ⁇ may be provided so as to weight to what extent the preceding iteration is taken into account.
- expression (10) is replaced by:
- P k 1 ⁇ ⁇ ( P k - 1 - P k - 1 ⁇ ⁇ k ⁇ ⁇ k T ⁇ P k - 1 ⁇ + ⁇ k T ⁇ P k - 1 ⁇ ⁇ k ) ( 12 )
- equation (12) is equivalent to equation (10).
- FIGS. 5B and 5C respectively, show, for the various potentials to which the light source 11 was raised, measurements of the reference intensity I ref (k) and the corrected values I′ ref (k) of the reference intensity resulting from the application of the recursive model without and with the weighting term taken into account.
- a model makes it possible to prevent fluctuations from affecting the reference intensity I ref .
- the recursive model allows a better approximation of the reference intensity to be obtained and adapts to non-linear variations in the intensity. Taking into account a weighting factor allows an optimal performance to be obtained. Specifically, in FIG.
- the present disclosure will possibly be implemented on processors for processing data measured by gas sensors, for applications in environmental monitoring, but also in applications related to the measurement of gas in industrial environments or in medical applications.
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Abstract
Description
-
- a) placing the gas between a light source and a measurement photodetector, the light source being able to emit an incident light wave, the incident light wave propagating through the gas toward the measurement photodetector, the measurement photodetector being able to detect a light wave transmitted by the gas, in the absorption spectral band;
- b) illuminating the gas with the light source;
- c) measuring, with the measurement photodetector, an intensity, called the measurement intensity, of the light wave transmitted by the gas, in the absorption spectral band; and
- d) measuring, with a reference photodetector, an intensity, called the reference intensity, of a light wave, called the reference light wave, the reference light wave being emitted by the light source;
-
- e) correcting the reference intensity detected at the various measurement times; and
- f) estimating an amount of the gaseous species, at the various measurement times, from the corrected reference intensity and from the measurement intensity;
-
- ei) taking into account a parametric model, defined using parameters, describing a temporal variation in the reference intensity detected at the various measurement times;
- eii) estimating the parameters of the model using the reference intensities measured at the various measurement times; and
- eiii) estimating a reference intensity called the denoised reference intensity at each measurement time depending on the estimated parameters of the model.
-
- in sub step ei), the model is determined for the time range;
- in sub step eii), the parameters of the model are estimated using reference measurements carried out during the time range; and
- in substep eiii), the reference light intensity is corrected for the measurement times of the time range.
-
- fi) from the corrected reference intensity, at each measurement time, estimating an intensity of the light wave emitted by the light source, in the absorption spectral band, at the measurement time;
- fii) comparing the intensity thus estimated with the measurement intensity resulting from step c), at the measurement time; and
- fiii) estimating the amount of the gaseous species depending on the comparison carried out in substep fii). This comparison may take the form of a ratio.
-
- a light source able to emit an incident light wave that propagates toward the gas, the incident light wave lying in an absorption spectral band of the gaseous species;
- a measurement photodetector able to detect a light wave transmitted by the gas, at various measurement times, and to measure an intensity thereof called the measurement intensity;
- a reference photodetector configured to measure an intensity, called the reference intensity, of a reference light wave emitted by the light source, at the various measurement times;
- a first processor, for computing a corrected reference intensity at the various measurement times, from the reference intensity measured by the reference photodetector at the measurement times, the first processor being configured to implement step e) of the method according to the first subject of the present disclosure; and
- a second processor, for estimating the amount of the gaseous species, at each measurement time, depending on the corrected reference intensity and the measurement intensity. The second processor may, in particular, be configured to implement step f) of the method according to the first subject of the present disclosure.
-
- a
light source 11, able to emit alight wave 12, called the incident light wave, so as to illuminate a gas G lying in the internal space; and - a
photodetector 20, called the measurement photodetector, able to detect alight wave 14 transmitted by the gas G, under the effect of the illumination of the latter by theincident light wave 12. Thelight wave 14 is referred to by the term measurement light wave.
- a
where:
-
- μ(cx(k)) is an attenuation coefficient dependent on the amount cx(k) at the time k;
- l is the thickness of gas passed through by the light wave in the
chamber 10; and - Ix(k) is the intensity of the incident light wave, at the time k, which corresponds to the intensity of the light wave, in the absorption spectral band Δx, reaching the
measurement photodetector 20 in the absence of absorbent gas in the chamber.
corresponds to an attenuation att(k) generated by the gaseous species in question at the time k.
-
- the
reference photodetector 20 ref is placed in a chamber isolated from the gas to be analyzed; in this case, the reference spectral band Δref may be none other than the absorption spectral band Δx; and - the
reference photodetector 20 ref is none other than themeasurement photodetector 20, a filter-adjusting means allowing the photodetector to be alternately associated with themeasurement filter 18 and with the referenceoptical filter 18 ref. It may, for example, be a question of a filter wheel. When the reference photodetector is none other than the measurement photodetector, the reference spectral band Δref is preferably separate from the absorption spectral band Δx.
- the
Iref,f being the reference intensity after application of a median filter to five successive samples.
-
- k corresponds to a time increment; and
- a and b are real numbers forming the vector of parameters θ.
Step 110: acquiring the measurements.
Φ is a matrix of (Nk, 2) size, with, in this example, Nk=K. The first column is formed by all of the time increments k in increasing order, the second column being formed from 1's.
Y ref=Φ·θ+ε (4), where:
-
- θ is the vector of parameters, of (2, 1) size, with
-
- ε is an error vector, of (Nk, 1) size.
I′ ref(k)=ak+b (6), where:
a and b are the terms of the vector {circumflex over (θ)}.
allows the amount cx(k) to be obtained as indicated above.
{circumflex over (θ)}=(ΦT W T WΦ)−1ΦT W T WY ref (7)
I ref(k)=a k k+b k. (8), where
-
- k corresponds to a time increment; and
- ak and bk are real numbers forming the vector of parameters
{circumflex over (θ)}k={circumflex over (θ)}k-1 +P kρk[y k−ρk T{circumflex over (θ)}k-1] (9)
where:
-
- ρk is a vector of [2, 1] size with
Each vector ρk is such that its transpose ρk T corresponds to the kth row of the observation matrix Φ described above with regard to step 130;
-
- yk is a scalar corresponding to the measurement Iref(k); and▪ Pk is a matrix of (2,2) size updated at each time increment k according to the expression:
In the first iteration (k=1), an initial matrix P0, with for example
is employed.
Pk=(Φk T·Φk)−1. Pk corresponds to the inverse of the autocorrelation matrix of the observation matrix Φk.
designating an initial measurement time.
-
- a
substep 222 of forming the vector ρk on the basis of the measurement Iref(k); - a
substep 224 of updating the matrix Pk by implementing expression (10) on the basis of the matrix Pk-1 obtained in a preceding iteration or, in the first iteration, of the initialized matrix P0; and - a
sub step 226 of estimating the vector of parameters {circumflex over (θ)}k with expression (9) based on the vector {circumflex over (θ)}k-1 obtained in a preceding iteration or, in the first iteration (k=1), on the initialized vector {circumflex over (θ)}0.
Step 230: Correcting the reference intensity.
- a
I′ ref(k)=ρk{circumflex over (θ)}k =a k k+b k. (11).
Step 240: Estimating the amount cx(k) of the gaseous species analyzed.
λ is a weighting term with λ∈]0,1]. For example λ=0.99.
Claims (10)
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